If a line is making same angle - with the positive direction of X-Y- and Z-axes- then show that -cos-11-3-

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Standard XII

Mathematics

Direction Cosines

If a line is ...

Question

If a line is making same angle $θ$ with the positive direction of $X -, Y -$ and $Z -$ axes, then show that $θ = {cos}^{- 1} (\frac{1}{\sqrt{3}}$ .

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Solution

Direction cosine of the line are
$l = cos θ, m = cos θ, n = cos θ$
$l^{2} + m^{2} + n^{2} = 1$
${cos}^{2} θ + {cos}^{2} θ + {cos}^{2} θ = 1$
$c o s^{2} θ = \frac{1}{3}$
$θ = {cos}^{- 1} (\frac{1}{\sqrt{3}})$

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If a line is making same angle θ with the positive direction of X−,Y− and Z−axes, then show that θ=cos−1(1√3.

Direction cosine of the line arel=cosθ,m=cosθ,n=cosθl2+m2+n2=1cos2θ+cos2θ+cos2θ=1cos2θ=13θ=cos−1(1√3)

If a line is making same angle $θ$ with the positive direction of $X -, Y -$ and $Z -$ axes, then show that $θ = {cos}^{- 1} (\frac{1}{\sqrt{3}}$ .

Direction cosine of the line are
$l = cos θ, m = cos θ, n = cos θ$
$l^{2} + m^{2} + n^{2} = 1$
${cos}^{2} θ + {cos}^{2} θ + {cos}^{2} θ = 1$
$c o s^{2} θ = \frac{1}{3}$
$θ = {cos}^{- 1} (\frac{1}{\sqrt{3}})$